# An object with a mass of 2 kg is acted on by two forces. The first is F_1= < 2 N , -7 N> and the second is F_2 = < 1 N, -6 N>. What is the object's rate and direction of acceleration?

Mar 28, 2017

The rate of acceleration is $= 6.67 m {s}^{-} 2$ in the direction of $= 103$º

#### Explanation:

The resultant force is

$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2}$

$= < 2 , - 7 > + < 1 , - 6 >$

$= < 3 , - 13 >$

We apply Newton's second Law

$\vec{F} = m \vec{a}$

Mass, $m = 2 k g$

$\vec{a} = \frac{1}{m} \cdot \vec{F}$

$= \frac{1}{2} < 3 , - 13 > = < \frac{3}{2} , - \frac{13}{2} >$

The magnitude of the acceleration is

$| | \vec{a} | | = | | < \frac{3}{2} , - \frac{13}{2} > | |$

$= \sqrt{{\left(\frac{3}{2}\right)}^{2} + {\left(\frac{13}{2}\right)}^{2}}$

$= \frac{\sqrt{178}}{2} = 6.67 m {s}^{-} 2$

The direction is $\theta = \arctan \left(\frac{- 13}{3}\right)$

The angle is in the 2nd quadrant

$\theta = \arctan \left(- \frac{13}{3}\right) = 180 - 77 = 103$º

Mar 28, 2017

$a = \frac{\sqrt{178}}{2} \text{ "m/s^2 " , } \alpha \cong - 89. {\overline{9}}^{o}$

#### Explanation:

${F}_{1} = < 2 N , - 7 N > \text{ "rArr" } {\vec{F}}_{1} = 2 i - 7 j$

${F}_{2} = < 1 N , - 6 N > \text{ "rArr" } {\vec{F}}_{2} = i - 6 j$

${\vec{F}}_{\text{net}} = \left(2 + 1\right) i + \left(- 7 - 6\right) j$

${\vec{F}}_{\text{net}} = 3 i - 13 j$

${F}_{\text{net}} = \sqrt{{3}^{2} + {\left(- 13\right)}^{2}}$

${F}_{\text{net}} = \sqrt{9 + 169}$

F_("net")=sqrt(178)" "N

$\text{acceleration : a}$

$\text{mass :m}$

a=F_("net")/m" The Newton's second law of motion"

$a = \frac{\sqrt{178}}{2} \text{ } \frac{m}{s} ^ 2$

$\tan \alpha = - \frac{13}{3}$

$\tan \alpha = - 4. \overline{3}$

$\alpha \cong - 89. {\overline{9}}^{o}$